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Math Help - Optimization Problem

  1. #1
    Junior Member asemh's Avatar
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    Optimization Problem

    Hi
    I was attempting an Optimization question but I just do not know how to start it.

    A man in a row boat, 5km from the nearest point on a straight shoreline, wishes to reach a point 12kms further down the shore. If he can row at the rate of 4kms per hour and run a rate of 6kms per hour, what course should he follow to reach his destination in the minimum time?

    I first started the answer by drawing a triangle and one co-ordinate for the man in the boat, another co-ordinate for the shoreline, and the last co-ordinate for the nearest point.

    How would i start attempting this question? I am not asking for just an answer, I want to know what my formula should start off like, and when I should derive, etc.

    Thanks in advance.
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  2. #2
    Senior Member apcalculus's Avatar
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    Draw another point between the destination and the nearest point on the shoreline. Suppose this point you just labeled is the point where the man should row to in order to optimize total time.

    A = boat
    B = nearest point on shoreline
    C = destination
    Z = point between B and C

    The total travel time is divided in two parts: time traveled in water and time traveled on land.

    Call AC = x, and try to express the sum of t_1 and t_2 in terms of x.

    Here is the diagram for a similar problem with different conditions:


    I hope this helps you get started on this question.

    Good luck!
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  3. #3
    Junior Member asemh's Avatar
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    Here is my try at making a diagram for the question.
    Please tell me if this is right or wrong.
    If its wrong how should i fix it?

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  4. #4
    MHF Contributor

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    Your diagram is fine- and I like the little boat! You can calculate the distance to be rowed,. as a function of x, by the Pythagorean formula of course.
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  5. #5
    Junior Member asemh's Avatar
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    Let me get this straight,
    So i am finding the Hypotenuse of the triange, Derive it and set it to 0=function, solve for x and then do (12-x) and then thats the final answer?

    Where does the point of him rowing and running come into place? Am i understanding something wrong here? Please inform me.

    Thanks
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